Some authors use the term exponential valuation rather than "valuation". In this case the term "valuation" means "absolute value".

A valuation v is called trivial (or the trivial valuation of K) if v(a) = 0 for all a in K×, otherwise it is called non-trivial.

For valuations used in geometric applications, the first property implies that any non-empty germ of an analytic variety near a point contains that point. The second property asserts that any valuation is a group homomorphism, while the third property is a translation of the triangle inequality from metric spaces to ordered groups.

It is possible to give a dual definition of the same concept using the multiplicative notation for Γ: if, instead of ∞, an element O[2] is given and the ordering and group law on Γ are extended by the rules

O ≤ α for all α in Γ,

O · α = α · O = O for all α in Γ,

then a valuation of K is any map

v : K → Γ ∪ {O}

satisfying the following properties for all a, b in K:

v(a) = O if, and only if, a = 0,

v(ab) = v(a) · v(b),

v(a + b) ≤ max(v(a), v(b)), with equality if v(a)≠v(b).

(Note that in this definition, the directions of the inequalities are reversed.)

A valuation is commonly assumed to be surjective, since many arguments used in ordinary mathematical research involving those objects use preimages of unspecified elements of the ordered group contained in its codomain. Also, the first definition of valuation given is more frequently encountered in ordinary mathematical research, thus it is the only one used in the following considerations and examples.

Two valuations v1 and v2 of K with valuation group Γ1 and Γ2, respectively, are said to be equivalent if there is an order-preserving group isomorphismφ : Γ1 → Γ2 such that v2(a) = φ(v1(a)) for all a in K×. This is an equivalence relation.

Two valuations of K are equivalent if, and only if, they have the same valuation ring.

Let L/K be a finite extension and let w be an extension of v to L. The index of Γv in Γw, e(w/v) = [Γw : Γv], is called the reduced ramification index of w over v. It satisfies e(w/v) ≤ [L : K] (the degree of the extension L/K). The relative degree of w over v is defined to be f(w/v) = [Rw/mw : Rv/mv] (the degree of the extension of residue fields). It is also less than or equal to the degree of L/K. When L/K is separable, the ramification index of w over v is defined to be e(w/v)pi, where pi is the inseparable degree of the extension Rw/mw over Rv/mv.

When the ordered abelian group Γ is the additive group of the integers, the associated valuation induces a metric on the field K. If K is complete with respect to this metric, then it is called a complete valued field. In general, a valuation induces a uniform structure on K, and K is called a complete valued field if it is complete as a uniform space. There is a related property known as spherical completeness: it is equivalent to completeness if Γ = Z, but stronger in general.

If π' is another irreducible element of R such that (π') = (π) (that is, they generate the same ideal in R), then the π-adic valuation and the π'-adic valuation are equal. Thus, the π-adic valuation can be called the P-adic valuation, where P = (π).

The previous example can be generalized to Dedekind domains. Let R be a Dedekind domain, K its field of fractions, and let P be a non-zero prime ideal of R. Then, the localization of R at P, denoted RP, is a principal ideal domain whose field of fractions is K. The construction of the previous section applied to the prime ideal PRP of RP yields the P-adic valuation of K.

Suppose that Γ is the set of non-negative real numbers. Then we say that the valuation is non-discrete if its range is not finite.

Suppose that X is a vector space over K and that A and B are subsets of X. Then we say that A absorbs B if there exists a α in K such that λ in K and |λ| ≥ |α| implies that B ⊆ λ A. A is called radial or absorbing if A absorbs every finite subset of X. Radial subsets of X are invariant under finite intersection. And A is called circled if λ in K and |λ| ≥ |α| implies λ A ⊆ A. The set of circled subsets of L is invariant under arbitrary intersections. The circled hull of A is the intersection of all circled subsets of X containing A.

Suppose that X and Y are vector spaces over a non-discrete valuation field K, let A ⊆ X, B ⊆ Y, and let f : X → Y be a linear map. If B is circled or radial then so is f−1(B){\displaystyle f^{-1}(B)}. If A is circled then so is f(A) but if A is radial then f(A) will be radial under the additional condition that f is surjective.